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1、Thin-PlateSplinesByLionelClark(ReferringtoDavidEberly’swork,http://www.geometrictools.com/)1.Thin-platesplinesDescriptionoftheProblemIndimensions,theideaofthin-platesplinesistochooseafunctionthatexactlyinterpolatessomedatapoints(whicharecalledcontrolp
2、oints),say,,andthatminimizesthebendingenergy,,whereistheHessianmatrixof(matrixofsecond-orderpartialderivativesof)andisthesumofsquaresofthematrixentries.Theinfinitesimalelementofhypervolumeis,wherearethecomponentsof.DescriptionoftheProblemOften,itisals
3、opossibletoformulatetheproblemwithasmoothingparameterforregularization.Afunctionischosenthatdoesnotnecessarilyexactlyinterpolateallthecontrolpointsbutthatdoesminimize.Thesmoothingparameterisandischosenapriori.Thesummationmakesitclearthattherearecontro
4、lpoints.SolutiontotheProblemFirstweconsidertheproblemwithoutsmoothingparameter.AfterapplyingthemethodofCalculusofVariations(afamousmethodinfunctionalanalysisforsearchingforextremevaluesofafunctional),thebiharmonicequationisderivedfrommultivariateEuler
5、-Lagrangeequation,whereistheLaplacianoperator,whichisappliedtwice.However,itisn’talwayspossiblethatthesolutiontothebiharmonicequationalsosatisfytheexactinterpolations.Sowehavetocompromisesomehow.SolutiontotheProblemThebiharmonicequationischosentocompr
6、omisesuchthatdon’tberequiredtovanisheverywherebutonlyneedtovanishwhereexceptatthecontrolpoints.Sothebiharmonicequationismodifiedtoanotherequation,say,,whichisanon-homogeneouslinearpartialdifferentialequation,whereisDiracdeltafunctionandareunknowncoeff
7、icients.SolutiontotheProblemTosolvetheaboveequation,wefirstsolveasimplerone,,towhichtheGreen’sfunctionisasolution.isaradialfunction,expressedexplicitlyaswhere,,.SolutiontotheProblemApplyingtheSuperpositionPrinciple,wegetthegeneralsolutionstothecomprom
8、isedequation,whereisthegeneralsolutionstothebiharmonicequation.SolutiontotheProblemAmong,weonlychoosethepolynomialfunctionsofdegreeone,becausetheirbendingenergyisnull.Sothesolutioniswherearethecomponentsofandareunknownparameters.SolutiontotheP
